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(Solved): The moment generating function of the random variable \( X \) is \[ M(t)=\frac{3}{3-t}, \] where \( ...
![The moment generating function of the random variable \( X \) is
\[
M(t)=\frac{3}{3-t},
\]
where \( t<3 \).
(i) Write the pdf](https://media.cheggcdn.com/study/c3f/c3fa9054-27ba-4a77-a7f5-5b87db239827/image)
The moment generating function of the random variable \( X \) is \[ M(t)=\frac{3}{3-t}, \] where \( t<3 \). (i) Write the pdf of \( X \). (ii) Write the mean and variance of \( X \). (iii) What is the probability that \( X \) is at most 2 ?
Expert Answer
(i) The pdf of an exponential variable(X) with mean 3 can be written as: f((x)={13e?x3x>00otherwise